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<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="group_1.html#G1" target="_self">HGrStr</a>(# <a href="numbers.html#K1">REAL</a> ,<a href="binop_2.html#K33">addreal</a>  #) as   <a href="group_1.html#NM1">Group</a> <b>by </b><i><a class="ref" href="group_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E2:85"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V7" target="_self">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" target="_self">GROUP_1:def 16</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:85"/>
 <a href="group_1.html#G1" target="_self">HGrStr</a>(# <a href="numbers.html#K1">REAL</a> ,<a href="binop_2.html#K33">addreal</a>  #) is  <a href="group_1.html#V7" target="_self">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>


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